# Explicit check of Ward identity (Peskin & Schroeder p. 160)

I am trying to check explicitly that the (Compton) amplitude

$$i\mathcal{M} = -ie^2\epsilon^*_\mu(k’)\epsilon_\nu(k)\bar u(p’)\left[\frac{\gamma^\mu \not k\gamma^\nu + 2\gamma^\mu p^\nu}{2p\cdot k}+\frac{-\gamma^\nu\not{k'}\gamma^\mu+2\gamma^\nu p ^\mu}{-2p\cdot k'}\right]u(p)\tag{5.74}$$

vanishes when we replace either $$\epsilon_\nu(k)$$ with $$k_\nu$$ or $$\epsilon^*_\mu(k’)$$ with $$k’_\mu$$. This would confirm the Ward identity, as suggested on page 160 of Peskin and Schroeder’s book “Introduction to QFT”.

However, with the replacement $$\epsilon_\nu(k) \rightarrow k_\nu$$, I obtain:

\begin{aligned}k_\nu\mathcal{M}^\nu(k) :&= -e^2\epsilon^*_\mu(k’)k_\nu\bar u(p’)\left[\frac{\gamma^\mu \not k\gamma^\nu + 2\gamma^\mu p^\nu}{2p\cdot k}+\frac{-\gamma^\nu\not{k'}\gamma^\mu+2\gamma^\nu p ^\mu}{-2p\cdot k'}\right]u(p) \\ &= -e^2\epsilon^*_\mu(k’)\bar u(p’)\left[\frac{\gamma^\mu (\not k)^2 + 2\gamma^\mu (p\cdot k)}{2p\cdot k}+\frac{-\not k\not{k'}\gamma^\mu+2\not k p ^\mu}{-2p\cdot k'}\right]u(p) \\ &= -e^2\epsilon^*_\mu(k’)\bar u(p’)\left[\frac{2\gamma^\mu (p\cdot k)}{2p\cdot k}+\frac{-\not k\not{k'}\gamma^\mu+2\not k p ^\mu}{-2p\cdot k'}\right]u(p) \\ &= -e^2\epsilon^*_\mu(k’)\bar u(p’)\left[\gamma^\mu+\frac{-\not k\not{k'}\gamma^\mu+2\not k p ^\mu}{-2p\cdot k'}\right]u(p)\end{aligned}

and the terms in the bracket apparently don’t cancel each other. Are there any further possible simplifications?

You can further simplify this expression by using the dirac equation $$0=(\not p-m)u(p)=\bar u(p')(\not p'-m)$$ and $$k+p=k'+p'$$. Then the second term can be expressed as $$2\not k p^\mu-\not k\not k'\gamma^\mu = 2(\not k'+\not p' - \not p)p^\mu -(\not k'+\not p' - \not p)\not k'\gamma^\mu\simeq 2\not k' p^\mu -(m-\not p)\not k'\gamma^\mu$$ The last equality holds only between the spinors. Then commuting $$\not p$$ through $$\not k' \gamma^\mu$$ gives $$\not p\not k'\gamma^\mu = 2pk'\gamma^\mu-\not k' \not p\gamma^\mu \simeq 2pk'\gamma^\mu - 2\not k' p^\mu + \not k'\gamma^\mu m$$ In total we then have $$2\not k p^\mu-\not k\not k'\gamma^\mu \simeq 2pk' \gamma^\mu$$ and the terms in the bracket cancel.